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Let $ \vec\Omega, \vec\Omega' $ be two unit vectors in $ \mathbb R^3 $ such that $ \vec\Omega\cdot\vec\Omega' = t $ and let $ r > 0 $. The multipole expansion for the exponential reads:

$$ \mathrm e^{\mathrm i r t} = 4\pi \sum_{l=0}^\infty \sum_{m=-l}^l \mathrm i^l j_l(r)Y_{lm}(\vec\Omega)Y^*_{lm}(\vec\Omega') \\ = \sum_{l=0}^\infty (2l+1)\mathrm i^l j_l(r) P_l(t). $$

In Mathematica this series is given by:

Sum[(2 l + 1) I^l SphericalBesselJ[l, r] LegendreP[l, t], {l, 0, ∞}]

However, it seems that Mathematica is unable to compute this and similar series, for example:

$$ \sum_{l=0}^\infty (2l+1) j_l^2(r) P_l(t) = \mathrm{sinc}(r\sqrt{2-2t}). $$

Is there any way to fix it?

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    $\begingroup$ Where do you have this identity from? $\endgroup$
    – Henrik Schumacher
    Sep 27, 2018 at 9:35
  • $\begingroup$ @Henrik, the first example is a special case of this identity. That being said, I do not recall Mathematica ever being equipped to handle these general sums. $\endgroup$
    – J. M.'s missing motivation ♦
    Sep 27, 2018 at 9:39
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    $\begingroup$ Mathematica is not a magic box that'll spit out a solution to any problem.Only WRI can fix that.But $\sum _{l=0}^{\infty } (2 l+1) j_l(r){}^2 P_l(t)=\text{sinc}\left(r \sqrt{1-t}\right)$ is not true $\endgroup$
    – Mariusz Iwaniuk
    Sep 27, 2018 at 11:16
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    $\begingroup$ You can check it: w[r_, t_] := Sum[(2 l + 1)*SphericalBesselJ[l, r]^2*LegendreP[l, t], {l, 0, 25}] // N; Plot3D[{Sinc[r*Sqrt[1 - t]], w[r, t]}, {r, 0, 2}, {t, 0, 2}] ??? $\endgroup$
    – Mariusz Iwaniuk
    Sep 27, 2018 at 12:16
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    $\begingroup$ @MariuszIwaniuk Thank you. A factor of 2 was missing. Now it is okay: w[r_, t_] := Sum[(2 l + 1)*SphericalBesselJ[l, r]^2*LegendreP[l, t], {l, 0, 25}] // N; Plot3D[{Sinc[r*Sqrt[2 - 2 t]], w[r, t]}, {r, 0, 2}, {t, 0, 2}] $\endgroup$
    – Appliqué
    Sep 27, 2018 at 12:26

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